Survey of JPEG compression history analysis

Transcription

1 Survey of JPEG compression history analysis Andrew B. Lewis Computer Laboratory Topics in security forensic signal analysis

2 References Neelamani et al.: JPEG compression history estimation for color images, IEEE Transactions on Image Processing 15(6), 2006 Hany Farid: Exposing digital forgeries from JPEG ghosts, IEEE Transactions on Information Forensics and Security 4(1), 2009 Andrew B. Lewis, Markus G. Kuhn: Exact JPEG recompression 1, to appear in SPIE Electronic Imaging: Visual Information Processing and Communication, Draft at

3 Outline Revision of JPEG compression/decompression algorithm Probability theory for parameter estimation JPEG compression history estimation for color images Exact JPEG recompression

4 The JPEG algorithm Parameters: Quantization: Q L (luma), Q C (chroma) Sub-sampling: 1 1 (luma), 2 2, 2 2 (chroma) also known as 4:2:0 sub-sampling Colour space: Y C b C r Image Colour space convert Y C b C r DCT Q L Encode 2 2 DCT Q C Encode 2 2 DCT Q C Encode

5 JPEG compression history estimation Probabilistically estimate settings used in the previous compression step Input: raw image in colour space F Output: compressed representation colour space G, sub-sampling scheme S and quantization tables Q {G, S, Q } = arg max P(Image, G, S, Q) G,S,Q = arg max P(Image G, S, Q)P(G, S, Q) G,S,Q

6 Terminology of inverse probability Unknown parameters θ, data D, assumptions H P(θ D, H) = P(D θ, H)P(θ H) P(D H) likelihood prior posterior = evidence The quantity of P(D θ, H) is a function of both D and θ. For fixed θ it defines a probability over D. For fixed D it defines the likelihood of θ. 2 2 More in David J. C. MacKay: Information Theory, Inference and Learning Algorithms, Cambridge University Press.

7 Maximum a posteriori estimation We wish to estimate θ on the basis of data D. The maximum likelihood (ML) estimate of the parameters from the data is ˆθ ML (D) = arg max P(D θ) θ and maximum a posteriori (MAP) estimate is ˆθ MAP (D) = arg max P(D θ)p(θ) θ

8 Expectation maximization ML estimate requires the marginal likelihood, if we have hidden variables. ˆθ ML (D) = arg max P(D θ) θ P(D θ) = Z P(D z, θ)p(z θ) Evaluating the sum is sometimes computationally infeasible. The expectation-maximization algorithm can be used instead. Expectation: Q(θ θ (t) ) = E Z x,θ (t)[log L(θ; x, Z)] Maximization: θ t+1 = arg max Q(θ θ (t) ) θ No guarantees of convergence

9 Interpolation characterisation as an expectation maximization problem Expectation: Q(θ θ (t) ) = E Z x,θ (t)[log L(θ; x, Z)] Maximization: θ t+1 = arg max Q(θ θ (t) ) θ x: the observed image samples f (x, y) θ: the interpolation kernel α and variance σ 2 Z: the p-map, an array of probabilities with the same dimensions as the image, where each probability indicates P(f (x, y) M 1 )

10 Compression history estimation as MAP problem ˆθ MAP (D) = arg max P( D θ )P( θ ) θ {G, S, Q } = arg max G, S, Q P( Image G, S, Q )P( G, S, Q )

11 Estimating quantization tables Q (1) Small set of possible values for G and S. C space to G, sub-sample with S, then forward DCT gives a near-periodic distribution of coefficients Ω G,S over image. G, S, Q and X G,S Ω G,S independent {G, S, Q } = arg max P(Ω G,S G, S, Q)P(G)P(S)P(Q) G,S,Q {G, S, Q } = arg max G,S,Q ex G,S Ω G,S P( X G,S G, S, Q)P(G)P(S)P(Q)

12 Estimating quantization tables Q (2) Since the decompressor s dequantization, DCT coefficients X q were IDCT ed; the results were up-sampled if necessary; and the image was converted to the RGB colour space. We received the image, and applied a forward colour space conversion; we downsampled the planes, if appropriate; and we applied the forward DCT to get e X. Rounding errors accumulate during every stage of this process. Therefore, we model the DCT coefficient values X = X q + Γ where original DCT coefficients X q are modelled by a sampled zero-mean Laplace distribution P(x) drawn from a truncated normal distribution x and rounding error Γ is P(x) x.

13 Estimating quantization tables Q (3) The Laplace distribution for DCT coefficient values has scale parameter λ, determined from the observations. The probability distribution of coefficient values after quantization/dequantization with factor q is P( X q = t q Z + ) = (k+0.5)q λ δ(t kq) exp ( λ τ ) dτ k Z (k 0.5)q 2 Rounding errors are independent, so we convolve this distribution with our error term s distribution and normalize: P( X = t q) P( X q = τ q)p(γ = t τ) dτ

14 Estimating quantization tables Q (4) If we assume a uniform prior P(q) for quantization factors, we can now find the most likely value for a particular quantization factor q = Qi,j by maximizing P( X q). This is repeated for each quantization factor Qi,j for (i, j) {(0, 0),..., (7, 7)}: q = arg max q Z + P( X q) ex Ω

15 Evaluation Allows for arbitrary colour space conversion, sub-sampling and quantization table parameters Not implementation specific Partially recovered quantization tables could be used to determine quality factor Quality factors q {1,..., 100} map onto quantization tables in many compressor implementations Statistical rather than exact Can t recover bitstream Tikhonov deconvolution filter introduces errors Errors in the quantization table when most DCT coefficients at a particular frequency are zero No data for the quantization table when all are zero (low quality factors)

16 Exact recompression Can we recover the original bitstream (including the quantization tables) when given the result of decompression? Due to rounding and mismatch between the compressor and decompressor operations, simplying invoking the compressor with the same parameters will not work. Can we provide a guarantee that if the provided image was produced by a particular JPEG decompressor, we will recover that bitstream? uncompressed image compression JPEG image decompressed image 1 decompression = recompression recompressed JPEG image(s) decompressed image 2 decompression

17 Applications of exact recompression The input to JPEG compressors is often previously compressed image data. Detecting this and recompressing exactly will reduce the information loss from recompression. Hinder forensic analysis double compression detection, JPEG ghosts,... Detect tampered regions in an uncompressed image, when the background was output by a JPEG decompressor Some copy-protection schemes rely on the fact that copies will be recompressed, lowering the quality.

18 Information loss in the decompressor Model sources of uncertainty (rounding, range limiting) when inverting operations in the decompressor using intervals of integers. We store intervals for all intermediate values in the decompressor. Because we are modelling the exact computations, exact recompressors are implementation-specific.

19 Chroma down-sampling example (1) IJG chroma upsampling filter weights contributions from neighbouring samples by 1 16 (1, 3, 3, 9) in order of increasing proximity v x,y = with weights 1 16 (8 + α w i 1,j 1 + β w i,j 1 + γ w i 1,j + δ w i,j ), (1, 3, 3, 9) x = 2i, y = 2j (3, 1, 9, 3) x = 2i 1, y = 2j (α, β, γ, δ) = (3, 9, 1, 3) x = 2i, y = 2j 1 (9, 3, 3, 1) x = 2i 1, y = 2j 1.

20 Chroma down-sampling example (2) We need to solve for the down-sampled weights w i,j : 1 v x,y = 16 (8 + α w i 1,j 1 + β w i,j 1 + γ w i 1,j + δ w i,j ) Interval arithmetic rules give [ 1 w i,j = δ (v x,y 16 (8 + α w i 1,j 1 + β w i,j 1 + γ w i 1,j )), ] 1 δ (v x,y (α w i 1,j 1 + β w i,j 1 + γ w i 1,j ))

21 Chroma down-sampling example (3) k 0 w x,y 0 [0, 255] at all positions 1 x w 2, 1 y h 2 repeat k k + 1 change scan order of (x, y) (,,, ) for each sample position (x, y) in the upsampled plane do for (i, j ) {(i 1, j 1), (i, j 1), (i 1, j), (i, j)} do w i,j s v x,y c ā : Equation satisfied with ā for w i,j, s for v x,y and current estimates w x,y for other w values. w i k,j w k 1 i,j w i,j end for end for until w k = w k 1 w x,y c w k

22 Performance Number of test images IJG quality factor infeasible blocks total blocks Figure: Recompression performance for a dataset of uncompressed images from the UCID colour images were compressed and decompressed at quality factors q {40, 60, 70, 80, 82, 84, 86, 88, 90}, then recompressed. The proportion of blocks at each quality factor which were not possible to recompress due to an infeasible search size is shown.